632 Dr. W. M. Thornton on the 
rate of penetration is therefore in this case about 3:3 inches 
a second. 
When the permeability 1s assumed constant and no core 
currents flow, equation 2 takes the well-known form 
Lb 4nSelo. 2. 
from which 
=i (1H. 1). a. 
If this curve be drawn with a mean value of L suitable to 
the circuit, the corresponding curve with core currents 
allowed to flow appears with a superposed wave. The 
solution of (4) is impossible unless w can be expressed as a 
function of time; and even if this were known, there would 
be a further complication in expressing the magnetic force as 
a function of the radius. The only course open is to find 
experimentally the curve of rise, and from it to deduce the 
changes of w. It is seen from figs. 3 and 4 that the relation 
of uw to t does not lend itself to simple treatment. 
The effect of the core currents may be summarized thus. 
When the core is of such dimensions or shape that the 
magnetizing wave from the circumference reaches the centre 
in a time equal to or less than the time-constant of the 
circuit, the rise is wavy; when it is greater the core currents 
merely prolong the time of orowth withont distorting the 
curve of rise. 
§ 3. The distribution and reaction of induced currents in 
solid cores present the most difficult and complex problems in 
applied magnetism. The regulation of alternators, the heating 
of magnet cores due to armature reaction, the sluggishness 
of response exhibited by machines with solid cores which 
depend upon series excitation, are examples. Most of the 
practical cases are beyond the reach of exact analysis; but in 
order to see whether Mr. Heaviside’s solution for the rise of 
magnetizing current In a coil surrounding a solid core gave 
any clue to the observed pause in the rate of rise when the 
core is very large, the following numerical illustration was 
worked out. The curves he gave * for a core 1 cm. radius 
are smooth and free from distortion. He expresses the 
magnetic distribution within a circular solid core, the magne- 
tizing coil of which has just been short-circuited, by a series 
* Electrical Papers, vol. i. p. 398. 
